We provide a detailed treatment of real-valued, smooth and bounded algebro-geometric solutions of the Camassa-Holm (CH) hierarchy and describe the associated isospectral torus. We employ Dubrovin-type equations for auxiliary divisors and certain aspects of direct and inverse spectral theory for self-adjoint Hamiltonian systems. In particular, we rely on Weyl-Titchmarsh theory for singular (cano...

We study the geodesic exponential maps corresponding to Sobolev type right-invariant (weak) Riemannian metrics μ (k ≥ 0) on the Virasoro group and show that for k ≥ 2, but not for k = 0, 1, each of them defines a smooth Fréchet chart of the identity. For k = 0 and k = 1 the corresponding geodesic flows are related to the Korteweg de Vries and Camassa Holm equations. In particular, the geodesic ...

We consider the Camassa-Holm equation with data in the energy norm H 1(R1). Global solutions are constructed by the small viscosity method for the frequency localized equations. The solutions are classical, unique and energy conservative. For finite band data, we show that global solutions for CH exist, satisfy the equation pointwise in time and satisfy the energy conservation law. We show that...

Using a Miura-Gardner-Kruskal type construction, we show that the Camassa-Holm equation has an infinite number of local conserved quantities. We explore the implications of these conserved quantities for global well-posedness.

In this paper we present a geometric interpretation of the periodic Degasperis-Procesi equation as the geodesic flow of a right invariant symmetric linear connection on the diffeomorphism group of the circle. We also show that for any evolution in the family of b-equations there is neither gain nor loss of the spatial regularity of solutions. This in turn allows us to view the Degasperis-Proces...

Reciprocal transformations [1-3] are a useful tool in the study of integrable partial differential equations. Recently, there has been much interest in the equation of Fokas-Fuchssteiner-CamassaHolm (FFCH), originally derived by Fokas and Fuchssteiner [4] in the context of bi-Hamiltonian theory, and subsequently rederived as an equation for shallow water waves by Camassa and Holm [5]. The FFCH ...

In this paper we present a geometric interpretation of the periodic Degasperis-Procesi equation as the geodesic flow of a right invariant symmetric linear connection on the diffeomorphism group of the circle. We also show that for any evolution in the family of b-equations there is neither gain nor loss of the spatial regularity of solutions. This in turn allows us to view the Degasperis-Proces...

We study the modulational instability of a shallow water model, with and without surface tension, which generalizes the Whitham equation to include bi-directional propagation. Without surface tension, the small amplitude periodic traveling waves are modulationally unstable if their wave number is greater than a critical wave number predicting a Benjamin-Feir type instability and the result qual...

We construct a new series of multicomponent integrable PDE systems that contain as particular example (with appropriately chosen parameters) many famous including KdV, coupled Harry Dym, Camassa-Holm, Kaup-Boussinesq systems. The contains also with no low-component analogues.

The Camassa-Holm (CH) and Hunter-Saxton (HS) equations have an interpretation as geodesic flow equations on the group of diffeomorphisms, preserving the H and Ḣ right-invariant metrics correspondingly. There is an analogy to the Euler equations in hydrodynamics, which describe geodesic flow for a right-invariant metric on the infinitedimensional group of diffeomorphisms preserving the volume el...